DOCSLIB.ORG

Pi Formulas, Ramanujan, and the Baby Monster Group By

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Pi Formulas, Ramanujan, and the Baby Monster Group By Pi Formulas, Ramanujan, and the Baby Monster Group By Titus Piezas III Keywords: Pi formulas, class polynomials, j-function, modular functions, Ramanujan, finite groups. Contents I. Introduction II. Pi Formulas A. The j-function and Hilbert Class Polynomials B. Weber Class Polynomials C. Ramanujan Class Polynomials III. Baby Monster Group IV. Conclusion I. Introduction In 1914, Ramanujan wrote a fascinating article in the Quarterly Journal of Pure and Applied Mathematics. The title was “Modular equations and approximations to p” and he discusses certain methods to derive exact and approximate evaluations. In fact, in his Notebooks, he gave several formulas, seventeen in all, for p (or to be more precise, 1/p). For a complete list, see http://mathworld.wolfram.com/PiFormulas.html. He gave little explanation on how he came up with them, other than saying that there were “corresponding theories”. One of the more famous formulas is given by, 1/(pÖ8) = 1/992 S r (26390n+1103)/3964n where r = (4n)!/(n!4) and the sum S (from this point of the article onwards) is to go from n = 0 to ¥. How did Ramanujan find such beautiful formulas? We can only speculate at the heuristics he used to find them, since they were only rigorously proven to be true in 1987 by the Borwein brothers. However, one thing we do know: Ramanujan wrote down in his Notebooks, among the others, the approximation1, epÖ58 » 24591257751.9999998222… which can also be stated as, epÖ58 » 3964 – 104 The appearance of 3964 in both the exact formula and the approximation is no coincidence. The connection can be highlighted even further when you realize that 26390 = 5*7*13*58. What are these numbers 58 and 3964? The former is a squarefree discriminant d of a quadratic form ax2+bxy+cy2 and 396 can be derived from the class invariant, 1 2 g58 = (5+Ö29)/2 2 2 which in turn is related to the fundamental solution (x1 = 5, y1 = 1) of the Pell equation x -29y = -4. How can a formula for p have something to do with Pell equations? That, I believe, is what makes a mathematical journey so rewarding: the unexpected and fascinating connections we stumble on along the way. We will not be able to give an answer to that question in this article but we can do so for a more tractable one: How do we derive 396 or, in general, other algebraic numbers involved in pi formulas, from class invariants? It turns out the answer has to do with transforming roots of Weber class polynomials into roots of Ramanujan class polynomials, polynomials which we have mentioned in “Ramanujan’s Constant (epÖ163) and its Cousins” [1]. In that article, we also discussed Hilbert class polynomials and pi formulas derived from them and that is where we will start. II. Pi Formulas A. The j-function and Hilbert Class Polynomials Similar to epÖ58 but more famous, is epÖ163 which can be shown to be approximately, epÖ163 » 6403203 + 743.99999999999925… The reason for such close approximation is because of the j-function, j(t), which has the q-series expansion, j(t) = 1/q + 744 + 196884q + 21493760q2 + 864299970q3 + … where q = e2pit and t is the half-period ratio defined as, Case 1: For d = 4m, t = Ö(-m) Case 2: For d = 4m+3, t = (1+Ö(-d))/2 where m is a positive integer, and d is the unsigned discriminant. In [1], we were quite liberal with our use of what we called the j-function but we will be more strict here. First, while in some older texts the convention was to label the j-function as j(q), it seems the modern one is to label it as j(t), or “j-tau”. Second, an obvious observation is that since d is positive, then t is imaginary. We are evaluating j(t) at imaginary arguments yet it will have a real value at those points. However, it can be positive or negative. For case 1, for all d > 0, then j(t) is always positive. For case 2, for d > 0, there are three cases, d < 3, j(t) is positive 2 d = 3, j(t) is zero d > 3, j(t) is negative From this point, what we will call as j(q) will refer to the absolute value of the j-function, while j(t) will refer to the signed value. This value is an algebraic integer of degree n where n is the class number of d. We have also observed that j(t) was a good approximation for numbers of the form epÖd. Case 1: If d = 4m, t = Ö(-m), then epÖ(4m) » j(t) – 744, for d > 12. Example 1. Let m = 5, d = 20 (class number 2), j(Ö-5) = 23(25+13Ö5)3 so, epÖ20 » 23(25+13Ö5)3 -744 Example 2. Let m = 10, d = 40 (class number 2), j(Ö-10) = 63(65+27Ö5)3 so, epÖ40 » 63(65+27Ö5)3 - 744 Case 2: If d = 4m+3, t = (1+Ö-d)/2, then epÖd » | j(t)| + 744, for d > 12. Example 1. Let d = 163 (class number 1), j((1+Ö-163)/2) = -6403203 so, epÖ163 » 6403203 + 744 Technically, case 2 is also defined for other d. For example , for d = 4m+2, say, d = 2, we have, j((1+Ö-2)/2) = 103(19-13Ö2)3 However, for the purposes of this article, we will limit case 2 only to form 4m+3. One should also take note that j(Ö-d) and j((1+Ö-d)/2) for d = 4m+3 are different, as illustrated by, j(Ö-7) = 2553, and j((1+Ö-7)/2) = -153 which can easily be confirmed by an adequate computer algebra system. Anyway, now that we have the preliminaries out of the way, we can go to the pi formulas. To recall, we pointed out that there are exactly nine discriminants d with class number 1, the absolute values given as 3, 4, 7, 8, 11, 19, 43, 67, 163, with the larger values giving the better approximations: 3 epÖ7 ˜ 153 + 697 epÖ11 ˜ 323 + 738 epÖ19 ˜ 963 + 744 epÖ43 ˜ 9603 + 744 epÖ67 ˜ 52803 + 744 epÖ163 ˜ 6403203 + 744 We also have the formulas due to the Chudnovsky brothers (which had an their inspiration Ramanujan’s formula cited in the opening paragraph): Let, c = (6n)!/((n!)3(3n)!), 1/(3p) = S c (-1)n (63n+8)/(153)n+1/2 1/(4p) = S c (-1)n(154n+15)/(323)n+1/2 1/(12p) = S c (-1)n (342n+25)/(963)n+1/2 1/(12p) = S c (-1)n (16254n+789)/(9603)n+1/2 1/(12p) = S c (-1)n (261702n+10177)/(52803)n+1/2 1/(12p) = S c (-1)n (545140134n+13591409)/(6403203)n+1/2 The formulas obviously use the j(q)’s of the given approximations. The factorization of a term in the numerator also indicate what d is involved, 63 = 32 *7 154 = 2*7*11 342 = 2*32 *19 16254 = 2*33 *7*43 261702 = 2*32 *7*31*67 545140134 = 2*32 *7*11*19*127*163 Now why is this? It turns out the answer is in the general form of the formula for d = 4m+3, d > 3, (implying j(t) as negative), 1/p = S c (-1)n (An+B)/(C)n+1/2 where, A = Ö(d*(1728-j(t))); C = -j(t) Since, 1728 = 123 (remember the taxicab anecdote?2) then, 7(123 + 153) = 72 *272 11(123 + 323) = 42 *112 *142 19(123 + 963) = 122 *192 *182 43(123 + 9603) = 122 *432 *3782 67(123 + 52803) = 122 *672 *39062 163(123 + 6403203) = 122 *1632 *33444182 For these d, the expression d*(1728-j(t)) is a perfect square! So there are beautiful Diophantine relationships behind these pi formulas as well, which explains why the factorization of A contains d. However, we are still missing the constant B and that is the fly in the ointment. 4 Technically, there is a closed-form expression for B in terms of Eisenstein series but it gets too complicated for this article. The interested reader is referred to “Ramanujan, Modular Equations, and Approximations to Pi” by Bailey, Borwein, and Borwein which contains a side article on the Chudnovskys’ approach as well as their own which we shall discuss later. The other way to find B (since A and C are already known) is simply to solve the equation, 1/p = S c (-1)n (An+B)/(C)n+1/2 using enough n terms which should give B to a reasonable accuracy. For these six d, B conveniently is an integer, given by 3*8, 4*15, 12*25, 12*789, 12*10177, and 12*13591409, respectively. Since A and B have a common factor, then one can see why 1/p is divided by either 3, 4, or 12. We have also pointed out that there are non-maximal or non-fundamental discriminants with class number 1 as shown by the approximations, epÖ16 ˜ 663 - 744 epÖ28 ˜ 2553 - 744 We can also use these two j(q)’s to find more pi formulas. The general form (with some small changes) for d = 4m is given by, 1/p = S c (An+B)/(C)n+1/2 where, A = Ö(-d*(1728-j(t))); C = j(t) Note that there is no (-1)n for d = 4m. The presence of (-1)n in the formula for d = 4m+3 is simply the negative sign of j(t). For d = 4m, since j(t) is positive, then there is no need for it.
Load more

Recommended publications
  • Moonshines for L2(11) and M12
    Moonshines for L2(11) and M12 Suresh Govindarajan Department of Physics Indian Institute of Technology Madras (Work done with my student Sutapa Samanta) Talk at the Workshop on Modular Forms and Black Holes @NISER, Bhubaneswar on January 13. 2017 Plan Introduction Some finite group theory Moonshine BKM Lie superalgebras Introduction Classification of Finite Simple Groups Every finite simple group is isomorphic to one of the following groups: (Source: Wikipedia) I A cyclic group with prime order; I An alternating group of degree at least 5; I A simple group of Lie type, including both I the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; I the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type). I The 26 sporadic simple groups. The classification was completed in 2004 when Aschbacher and Smith filled the last gap (`the quasi-thin case') in the proof. Fun Reading: Symmetry and the Monster by Mark Ronan The sporadic simple groups I the Mathieu groups: M11, M12, M22, M23, M24; (found in 1861) I the Janko groups: J1, J2, J3, J4; (others 1965-1980) I the Conway groups; Co1, Co2, Co3; 0 I the Fischer groups; Fi22. Fi23, Fi24; I the Higman-Sims group; HS I the McLaughlin group: McL I the Held group: He; I the Rudvalis group Ru; I the Suzuki sporadic group: Suz; 0 I the O'Nan group: O N; I Harada-Norton group: HN; I the Lyons group: Ly; I the Thompson group: Th; I the baby Monster group: B and Sources: Wikipedia and Mark Ronan I the Fischer-Griess Monster group: M Monstrous Moonshine Conjectures I The j-function has the followed q-series: (q = exp(2πiτ)) j(τ)−744 = q−1+[196883+1] q+[21296876+196883+1] q2+··· I McKay observed that 196883 and 21296876 are the dimensions of the two smallest irreps of the Monster group.
    [Show full text]
  • The Logbook of Pointed Hopf Algebras Over the Sporadic Groups
    THE LOGBOOK OF POINTED HOPF ALGEBRAS OVER THE SPORADIC SIMPLE GROUPS N. ANDRUSKIEWITSCH, F. FANTINO, M. GRANA˜ AND L. VENDRAMIN Contents Introduction 1 1. Algorithms 2 1.1. Algorithms for type D 2 1.2. Structure constants 4 1.3. Bases for permutation groups 4 1.4. An algorithm for involutions 4 2. Some auxiliary groups 4 2.1. The groups A9, A11, A12, S12 5 2.2. The group L5(2) 5 2.3. The group O7(3) 5 + 2.4. The group O8 (2) 5 − 2.5. The group O10(2) 6 2.6. The exceptional group G2(4) 6 2.7. The exceptional groups G2(3) and G2(5) 7 2.8. The symplectic group S6(2) 7 2.9. The symplectic group S8(2) 7 2.10. TheautomorphismgroupoftheTitsgroup 7 2.11. Direct products 8 3. Proof of Theorem II 8 3.1. The Lyons group Ly 8 3.2. The Thompson group Th 10 3.3. The Janko group J4 11 3.4. The Fischer group Fi23 12 3.5. The Conway group Co1 13 3.6. The Harada-Norton group HN 14 ′ 3.7. The Fischer group Fi24 14 3.8. The Baby Monster group B 15 arXiv:1001.1113v2 [math.QA] 25 Oct 2010 3.9. The Monster group M 17 Appendix.Realandquasi-realconjugacyclasses 18 References 21 Introduction In these notes, we give details on the proofs performed with GAP of the theorems of our paper [AFGV2]. Here we discuss the algorithms implemented for studying Date: October 31, 2018. 2000 Mathematics Subject Classification. 16W30; 17B37.
    [Show full text]
  • Words for Maximal Subgroups of Fi '
    Open Chem., 2019; 17: 1491–1500 Research Article Faisal Yasin, Adeel Farooq, Chahn Yong Jung* Words for maximal Subgroups of Fi24‘ https://doi.org/10.1515/chem-2019-0156 received October 16, 2018; accepted December 1, 2019. energy levels, and even bond order to name a few can be found, all without rigorous calculations [2]. The fact that Abstract: Group Theory is the mathematical application so many important physical aspects can be derived from of symmetry to an object to obtain knowledge of its symmetry is a very profound statement and this is what physical properties. The symmetry of a molecule provides makes group theory so powerful [3,4]. us with the various information, such as - orbitals energy The allocated point groups would then be able to levels, orbitals symmetries, type of transitions than can be utilized to decide physical properties, (for example, occur between energy levels, even bond order, all that concoction extremity and chirality), spectroscopic without rigorous calculations. The fact that so many properties (especially valuable for Raman spectroscopy, important physical aspects can be derived from symmetry infrared spectroscopy, round dichroism spectroscopy, is a very profound statement and this is what makes mangnatic dichroism spectroscopy, UV/Vis spectroscopy, group theory so powerful. In group theory, a finite group and fluorescence spectroscopy), and to build sub-atomic is a mathematical group with a finite number of elements. orbitals. Sub-atomic symmetry is in charge of numerous A group is a set of elements together with an operation physical and spectroscopic properties of compounds and which associates, to each ordered pair of elements, an gives important data about how chemical reaction happen.
    [Show full text]
  • On the Action of the Sporadic Simple Baby Monster Group on Its Conjugacy Class 2B
    London Mathematical Society ISSN 1461–1570 ON THE ACTION OF THE SPORADIC SIMPLE BABY MONSTER GROUP ON ITS CONJUGACY CLASS 2B JURGEN¨ MULLER¨ Abstract We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity- free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element 1+22 is a maximal subgroup of shape 2 .Co2.Thisisoneofthe first applications of a new general computational technique to enumerate big orbits. 1. Introduction The aim of the present work is to determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the Baby Monster group B, the second largest of the sporadic simple groups, on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup 1+22 of shape 2 .Co2. The final result is given in Table 4. In general, the endomorphism ring of a permutation module reflects aspects of the representation theory of the underlying group. Its character table in particular encodes information about the spectral properties of the orbital graphs associated with the permutation action, such as distance-transitivity or distance-regularity, see [14, 5], or the Ramanujan property, see [7]. Here multiplicity-free actions, that is those whose associated endomorphism ring is commutative, have been of particular interest; for example a distance-transitive graph necessarily is an orbital graph associated with a multiplicity-free action. The multiplicity-free permutation actions of the sporadic simple groups and the related almost quasi-simple groups have been classified in [3, 16, 2], and the as- sociated character tables including the one presented here have been collected in [4, 20].
    [Show full text]
  • Verification of the Ordinary Character Table of the Baby Monster 3
    VERIFICATION OF THE ORDINARY CHARACTER TABLE OF THE BABY MONSTER THOMAS BREUER, KAY MAGAARD, AND ROBERT A. WILSON Abstract. We prove the correctness of the character table of the sporadic simple Baby Monster group that is shown in the ATLAS of Finite Groups. MSC: 20C15,20C40,20D08 In memory of our friend and colleague Kay Magaard, who sadly passed away during the preparation of this paper. 1. Introduction Jean-Pierre Serre has raised the question of verification of the ordinary character tables that are shown in the ATLAS of Finite Groups [6]. This question was partially answered in the paper [5], the remaining open cases being the largest two sporadic simple groups, the Baby Monster group B and the Monster Group M, and the double cover 2.B of B. The current paper describes a verification of the character table of B. The computations shown in [3] then imply that also the ATLAS character table of 2.B is correct. As in [5], one of our aims is to provide the necessary data in a way that makes it easy to reproduce our computations. The ATLAS character table of the Baby Monster derives from the original cal- culation of the conjugacy classes and rational character table by David Hunt, de- scribed very briefly in [9]. The irrationalities were calculated by the CAS team in Aachen [11]. 2. Strategy We begin with a preliminary section, Section 3, whose aim is to prove that certain specified matrices do indeed generate copies of the Baby Monster. These arXiv:1902.07758v2 [math.RT] 20 May 2019 matrices can then be used in the main computation.
    [Show full text]
  • The Topological Index and Automorphism Group of 1,3,5
    Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM) Vol. 11, Issue 2 (December 2016), pp. 930 - 942 Study on the Q-Conjugacy Relations for the Janko Groups Ali Moghani Department of Computer Science William Paterson University Wayne, NJ 07470 USA E-mail: [email protected] Received March 5, 2016; Accepted July 22, 2016 ABSTRACT In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured. Keywords: Finite group; Sporadic, Janko; Conjugacy class; Character, Q-conjugacy; matured MSC 2010: 20D99, 20C15 1. Introduction In recent years, group theory has drawn wide attention of researchers in mathematics, physics and chemistry, see Fujita (1998). Many problems of the computational group theory have been researched, such as the classification, the symmetry, the topological cycle index, etc. They include not only the diverse properties of finite groups, but also their wide-ranging connections with many applied sciences, such as Nanoscience, Chemical Physics and Quantum Chemistry. For instance, see Darafsheh et al.
    [Show full text]
  • Arxiv:1610.03625V2 [Math.GR]
    THE FSZ PROPERTIES OF SPORADIC SIMPLE GROUPS MARC KEILBERG Abstract. We investigate a possible connection between the FSZ properties of a group and its Sylow subgroups. We show that the simple groups G2(5) 6 and S6(5), as well as all sporadic simple groups with order divisible by 5 are not FSZ, and that neither are their Sylow 5-subgroups. The groups G2(5) and HN were previously established as non-FSZ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be FSZ. We conclude by considering all perfect groups available through GAP with order at most 106, and show they are non-FSZ if and only if their Sylow 5-subgroups are non-FSZ. Introduction The FSZ properties for groups, as introduced by Iovanov et al. [4], arise from considerations of certain invariants of the representation categories of semisimple Hopf algebras known as higher Frobenius-Schur indicators [5, 10, 11]. See [9] for a detailed discussion of the many important uses and generalizations of these in- variants. When applied to Drinfeld doubles of finite groups, these invariants are described entirely in group theoretical terms, and are in particular invariants of the group itself. The FSZ property is then concerned with whether or not these invariants are always integers—which gives the Z in FSZ. While the FSZ and non-FSZ group properties are well-behaved with respect to direct products [4, Example 4.5], there is currently little reason to suspect a partic- ularly strong connection to proper subgroups which are not direct factors.
    [Show full text]
  • The Intersection Graph of a Finite Simple Group Has Diameter at Most 5
    The intersection graph of a finite simple group has diameter at most 5 Saul D. Freedman Abstract. Let G be a non-abelian finite simple group. In addition, let ∆G be the intersection graph of G, whose vertices are the proper nontrivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of ∆G has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension. Mathematics Subject Classification (2010). Primary 05C25; Sec- ondary 20E32. Keywords. Intersection graph, simple group, subgroups. arXiv:2009.02884v2 [math.GR] 28 Jan 2021 1. Introduction For a finite group G, let ∆G be the intersection graph of G. This is the graph whose vertices are the proper nontrivial subgroups of G, with two distinct vertices S1 and S2 joined by an edge if and only if S1 ∩ S2 6= 1. We write d(S1,S2) to denote the distance in ∆G between vertices S1 and S2, and if these vertices are joined by an edge, then we write S1 ∼ S2. Additionally, diam(∆G) denotes the diameter of ∆G. Cs´ak´any and Poll´ak [5] introduced the graph ∆G in 1969, as an analogue of the intersection graph of a semigroup defined by Bos´ak [1] in 1964. For finite non-simple groups G, Cs´ak´any and Poll´ak determined the cases where The author was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews.
    [Show full text]
  • An Introduction to Finite Simple Groups
    An Introduction to Finite Simple Groups Bryan Crompton Abstract Algebra Fall 2010 University of Massachussets Lowell Recall Some Definitions A set G with a binary operation is a group if (1) associativity holds, (2) the identity property holds, (3) and the inverse property holds. A subgroup H of a group G is a subset of G and also a group. A normal subgroup H of a group G is a subgroup such that gHg −1 = H for all g 2 H. A group is called simple if its only normal subgroups are f1g and itself. This doesn't seem like a lot to ask, but trying to understand what all simple groups there are leads to fascinating complexities! Who cares about finite simple groups? I do! But... •There are only two possible homomorphisms from a simple group. Since ker f is normal in G, and it can only be f1g or G, any homomorphic image is G or f1g by the First Isomorphic Theorem. So these groups are simple. • They are like primes and irreducible polynomials, but instead are fundamental building blocks for groups. • Given all finite simple groups, every other finite group can be generated. • There is a fascinating classification theorem! Jordan-Holder Theorem for Finite Groups Let G be a group and consider a sequence of normal groups G = G0 B G1 B G2 B ··· B Gr−1 B Gr = f1g called a composition series and where each quotient group Gi+1=Gi , called a composition factor, is simple. Then the theorem states that given a group, there is a unique (unordered) set of composition factors.
    [Show full text]
  • Supersingular Elliptic Curves and Moonshine
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 007, 17 pages Supersingular Elliptic Curves and Moonshine Victor Manuel ARICHETA yz y Department of Mathematics, Emory University, Atlanta, GA 30322, USA E-mail: [email protected] z Institute of Mathematics, University of the Philippines, Diliman 1101, Quezon City, Philippines E-mail: [email protected] Received September 30, 2018, in final form January 19, 2019; Published online January 29, 2019 https://doi.org/10.3842/SIGMA.2019.007 Abstract. We generalize a theorem of Ogg on supersingular j-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure. Key words: moonshine; modular curves; supersingular elliptic curves; supersingular polyno- mials 2010 Mathematics Subject Classification: 14H52; 11F06; 11F11; 11F22; 11F37; 20D08 1 Introduction and results Moonshine refers to unexpected connections between disparate areas of mathematics (e.g., mod- ular objects, sporadic groups) and physics (e.g., 2-dimensional conformal field theories).
    [Show full text]
  • Loop Groups and Noncommutative Geometry
    LOOP GROUPS AND NONCOMMUTATIVE GEOMETRY SEBASTIANO CARPI AND ROBIN HILLIER Abstract. We describe the representation theory of loop groups in terms of K- theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level ` projective unitary positive-energy representations of any given loop group L G. The construction is based on certain supersymmetric conformal field theory models associated with L G in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net. 1. Introduction Since its foundation by Connes [20], noncommutative geometry has been of growing importance, with impact on various fields of mathematics and physics: differential geometry, algebraic topology, index theory, quantum field theory, quantum Hall effect descriptions, etc., to name a few. The core idea is to work with an algebraic approach, which is in some sense \dual" to a topological one, namely instead of a given locally compact topological space one works with the commutative C*-algebra of continuous functions on it vanishing at infinity. It turns out that this offers several different and useful tools. According to Gelfand-Neimark's theorem, every commutative C*- algebra is actually of this type. To make things meaningful, usually some additional structure is requested to be given alongside the C*-algebra of continuous functions. The step to the noncommutative setting consists now basically in still requiring this further structure but allowing for arbitrary noncommutative C*-algebras (or even more general noncommutative topological algebras) instead of the commutative C*-algebra of continuous functions (cf.
    [Show full text]
  • Conjugacy Class Representatives in Fischer's Baby Monster
    Conjugacy class representatives in Fischer's Baby Monster Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham B15 2TT publsihed in LMS J. Comput. Math. 5 (2002), 175{180 Abstract We determine a set of conjugacy class representatives for the ele- ments in Fischer's Baby Monster simple group, up to algebraic conju- gacy. 1 Introduction Fischer's Baby Monster group is the second largest of the 26 sporadic simple groups, and has order greater than 4×1033. Many of its basic properties were described by Fischer [3], and its character table was computed by Hunt [4] (see also [2]). It was first constructed by Leon and Sims [5], essentially as a permutation group on some 1010 points. In [12] the author constructed the 4370-dimensional representation of the Baby Monster over GF (2). In [8] the 4371-dimensional representation was constructed over GF (3) and GF (5). The method could in principle be ap- plied also to construct the representation over any field of odd characteristic. These matrix constructions now give almost enough invariants to distin- guish all conjugacy classes of elements. In this paper we use this to produce a complete set of conjugacy class representatives for the Baby Monster. 1 Table 1: The elements a to z a b c = ab d = cb e = cd f = ce g = fc h = gd 2 3 55 55 40 20 12 18 i = ch j = id k = jd l = ck m = lc n = md o = nd p = eo 31 23 17 46 16 46 24 34 q = pe r = qc s = dr t = sc u = no v = il w = gh 36 60 26 47 47 34 47 x = (a(cg)24)6 y = ((cg)6x)12 z = ((cg)4xcgx)15 2 2 2 As usual, we note that it is sufficient to find representatives for the max- imal cyclic subgroups, as conjugacy class representatives for the elements can then be found as suitable powers of the given generators of the maximal cyclic subgroups.
    [Show full text]